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Full text of "Treatise On Analysis Vol-Ii"

360 XV NORMED ALGEBRAS AND SPECTRAL THEORY (15.8.1) Let b be a closed left ideal in A, and put Ub(x) - y = xy for all y e b and all x e A. Then x\-> Ub(x) is a representation of the algebra A on the Hilbert space b. First of all, Ub(x) is a continuous operator on b, by (15.7.5.3). Clearly Ub(xx') = Ub(x)Ub(xf). For y, z in b, we have (Ub(x)*-y\z) = (y\Ub(x) • z) = (y\xz) = (x*y\z) = (Ub(x*)-y\z) by virtue of (15.7.5.2), hence Ub(x)* = Ub(x*). Also if A has a unit element e, then Ub(e) is the identity transformation. When b = A, we write U(x) in place of UA(x); the representation U is called the regular representation of A. It is faithful (1 5.5) by virtue of (1 5.7.5.7). Moreover, x\-+ U(x) is a continuous linear mapping of A into =2?(A), by the continuity of (x, y) i-* xy. The study of complete Hilbert algebras is founded on the consideration of their minimal left ideals and the idempotents which generate them. For each left ideal I of A, we denote by I* the image of I under the involu- tion ^H*,?*; clearly I* is a right ideal. (15.8.2) For each left ideal I of A, the orthogonal supplement I1 of the closure I 0/1 is a left ideal. Since I is a left ideal (15.1.3), we may as well assume that I is closed. If y e I1 and z e A, then for each x e I we have (zy \ x) = (y \ z*x) = 0 because I is a left ideal, whence it follows that zy e I1. (15.8.3) Lete^Q be an idempotent in A. Then (i) Ikll^i; (ii) e* is idempotent ; (iii) the left ideal Ae is the set of all x e A such that x = xe, and is closed in A. The first assertion follows from the inequality ||i|| = \\e2\\ ^ \\e\\2. The second is trivial. As to (iii), it is clear that if x = xe then x e Ae, and conversely if x e Ae, then x = ye for some y e A, hence xe = ye2 = ye = x. The fact that Ae is closed then follows from the continuity of the mapping xt~+x — xe, and (3.1 5.1). Consider in particular the self-adjoint idempotents (cf. (15.5.3.1 )): (15.8.4) Ifel9 e2 are self-adjoint idempotents in A, then the following proper- ties are equivalent: (a) (e± \e2) = 0; (b) e±e2 == 0; (c) e2 et = 0.te Hilbert algebra.